If f(z) is analytic within and on a closed curve C and a is a point within C, then
f(a) = (1/2πi)∮_{c} [f(z)/(za)] dz
Cauchy's Integral Formula for the derivative of an analytic function
If a function f(z) is analytic in a region D, then its derivative at any point z=a of D is also analytic in D and is given by
f '(a) = (1/2πi)∮_{c} [f(z)/(za)^{2}] dz
where C is any closed contour in D surrounding the point z=a.
Theorem
If a function f(z) is analytic in a region D, then its derivative at any point z=a of D has derivatives of all orders, all of which are again analytic function in D, their values are given by
f^{n}(a) = (n!/2πi)∮_{c} [f(z)/(za)^{n+1}] dz
where C is any closed contour in D surrounding the point z=a.
